We finalize this section by presenting a deep operator network [1]}, [2]} that learns the geodesic distance between two points on a manifold. This type of networks learn to transform functions instead of single points. They are composed by a branch network that encodes the input functions and a trunk net that encodes the input position. The input function in our case corresponds to boundary condition of the Eikonal equation, shown in equation REF  and the input coordinate corresponds to points in the manifold. We note two properties of the geodesic distance function \(d(\cdot , \cdot )\) : first, it is symmetric, \(d(\mbox{$x$}{}_1, \mbox{$x$}{}_2) = d(\mbox{$x$}{}_2, \mbox{$x$}{}_1)\) , since the distance from \(\mbox{$x$}{}_1\)  to \(\mbox{$x$}{}_2\)  is the same as the distance from \(\mbox{$x$}{}_2\)  to \(\mbox{$x$}{}_1\) . Second, the distance from a point to itself is zero, that is \(d(\mbox{$x$}{}, \mbox{$x$}{}) = 0\)  for every point in the manifold. As in the rest of the experiments, we will represent points in the manifold by the eigenfunctions of the Laplace-Beltrami operator \(\mbox{$v$}{}_i = [v_1(\mbox{$x$}{}_i), v_2(\mbox{$x$}{}_i),...,v_N(\mbox{$x$}{}_i)]\) . With this is mind, we propose the following deep operator network architecture \(\hat{d}\)  to approximate \(d\) :
\(\bar{d}(\mbox{$v$}{}_1,\mbox{$v$}{}_2) &=& NN(\mbox{$v$}{}_1;\mbox{$\theta $}{}_T) \cdot NN(\mbox{$v$}{}_2;\mbox{$\theta $}{}_B)\\\hat{d}(\mbox{$v$}{}_1,\mbox{$v$}{}_2) &=& \frac{1}{2}\left( 1 - \frac{\mbox{$v$}{}_1 \cdot \mbox{$v$}{}_2}{\Vert  \mbox{$v$}{}_1\Vert  \Vert  \mbox{$v$}{}_2\Vert } \right)(\bar{d}(\mbox{$v$}{}_1,\mbox{$v$}{}_2) + \bar{d}(\mbox{$v$}{}_2,\mbox{$v$}{}_1))\) 
